Explicit Computations of Fundamental Classes

TL;DR

This paper employs group cohomology techniques to explicitly compute fundamental classes and applies these methods to determine the Tate canonical class for certain cyclotomic extensions.

Contribution

It provides explicit calculations of fundamental classes using cohomological methods and applies these to compute the Tate canonical class for specific number field extensions.

Findings

Explicit formulas for local fundamental classes

Method to compute Tate canonical class for cyclotomic extensions

Enhanced understanding of cohomological computations in number theory

Abstract

We use the techniques of group cohomology to give explicit computations of the local fundamental class. As an application, we discuss how to compute the Tate canonical class for the extension $\mathbb Q(\zeta_{p^\nu})/\mathbb Q$, where $p^\nu$ is an odd prime power.